If \(f\)  is a polynomial then \(\mathcal {I}(f)\)  is simply the basin of (immediate) attraction to infinity which is a central object in the study of the dynamics of polynomials.
If \(f\in \mathcal {B}\)  is transcendental then \(\mathcal {I}(f)\ne \emptyset \)  and \(\mathcal {I}(f)\subset \mathcal {J}(f)\)  (see [1]} and [2]}) and we are interested in
\(\text{{\rm HD}}(\mathcal {I}(f)),\) 
